First slide

Introduction to integration

Question

If the antiderivative of $f (x) =$
$\frac{\sin x}{\sin^{2} x + 4 \cos^{2} x} is (1 / \sqrt{3}) \tan^{- 1} ((1 / \sqrt{3}) g (x)) + C$ then
$g (x)$ is equal to

Easy

A

$\sec x$
B

$\tan x$
C

$\sin x$
D

$\cos x$

Solution

$\begin{array}{l} f (x) = \frac{\sin x}{\sin^{2} x + 4 \cos^{2} x} = \frac{\tan x \sec x}{\tan^{2} x + 4} \\ = \frac{\tan x \sec x}{\sec^{2} x + 3} \end{array}$
Putting $\sec x = t, d x \sec x \tan x = d t$ so
$\int f (x) d x = \int \frac{d t}{t^{2} + 3} = \frac{1}{\sqrt{3}} \tan^{- 1} \frac{\sec x}{\sqrt{3}} + C$

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